(Pólya’s Urn) An urn initially contains \(r\) red and \(b\) blue balls. A ball is chosen uniformly at random (i.e. with probability \(1/(r+b)\) each). If If it comes up red (resp. blue), then it is returned and another red (resp. blue) ball is added to the urn. The process is repeated indefinitely. Let \(R_n\) be the number of red balls in the urn after \(n\) draws.
a) Prove that the fraction of red balls \(M_n=R_n/(r+b+n)\) has an almost sure limit \(M_\infty \) as \(n\to \infty \).
b) Suppose now \(r=b=1\). Prove that \(\mathbb {P}(R_n=k)=1/(n+1)\) for \(1\leq k\leq n+1\). Hint: Write down the probability of choosing \(k\) red and \(n-k\) blue balls in some fixed order.
c) Still assuming \(r=b=1\), what is the distribution of \(M_\infty \)? Hint: use part b) to compute \(\mathbb {P}(M_\infty \leq x)\).